Advanced Microeconomic Theory Fall 2014 Hannu Vartiainen Problem Set 3 1. Argue that, in the Second Wefare Theorem, if the consumers’preferences are strictly convex, the Pareto-e¢ cient x is the unique Walrasian allocation that results from the initial endowment x: 2. Consider n agent exchange economy with n consumers with identical and strictly concave utility functions. Let there be some initial endowment vector !: Show that equal division is a Pareto-e¢ cient allocation. 3. Consider one consumer, one …rm economy (Robinson Crusoe) where the consumer’s utility function is of the Cobb-Douglas form u(x; `) = x`; where x is is the consumption good and ` is the amount of leisure. The …rm’s production function is linear x = y ; where y is the labor inputand 2 (0; 1). The resource constraint on the amount of leisure and labor is ` + y = 1: The consumer owns the …rm. (a) Write down the consumer’s and the …rms optimization problems, given the prices p of the output and w of the labor input. (b) Find the demand function of the consumer and the production function of the …rm (as a function of p and w) (c) Solve the Walrasian equilibrium prices p and w (d) Compute the equilibrium utility of the consumer. What happens to the utility when changes? 4. Suppose that in the previous exercise, the consumer is an entrepreneur who participates a large market, and adjusts her production and consumption to the exogenously given prices p0 and w0 . Show that he utility does not decrease relative to the Robinson Crusoe case. 5. Let X = fx1 ; x2 ; x3 g be the set of pure outcomes where vNM preferences % satisfy x1 x2 x3 . A lottery 1k o¤ers xk with certainty. Show that there is 2 [0; 1] such that 12 11 + (1 ) 13 : 1 6. Let a risk averse agent with Bernoulli utility function u( ) decide his insurance coverage x > 0. The probability of an accident is > 0 and the corresponding montery loss is L: Coverage x re‡ects the compensation from the insuance company in the case of an accident. Let the constant marginal cost of insurance be exactly : What coverage should the agent choose? 7. By using the formal de…nition of risk-aversion, show that an agent is risk-averse has a strictly concave Bernoulli utility function. 8. Argue that if a risk averse decision maker rejects a …xed favorable bet, i.e. one whose expected value is larger than the cost of participating the bet, at all levels of initial wealth, then the Bernoulli utility of the decision maker is bounded from above. 2

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